We here attempt to give a more consistent view on these issues and, arguably, a more accurate depiction.
In many cases it is a too crude assumption to presume that while \(y\)  is deranged by noise, \(x\)  is not. Standard estimation procedures such as maximum likelihood or nonlinear least-squares become biased for a regression model of the form
(REF ) when \(x\)  is observed with noise [1]}. Furthermore, the separate quantification of the aleatoric part of the uncertainty due to the noise of the input is not possible in a model like (REF ) since the variance \(\sigma _y^2\)  also accounts for noise in the output.
Following the idea of Errors-in-Variables (EiV), a quite classical concept in statistics [1]}, we try to resolve these issues by changing (REF ) to
\(\begin{aligned}x &= \zeta + \varepsilon _x \,,\qquad y = f_\theta (\zeta ) + \varepsilon _y \,,\end{aligned}\) 
